= Quadratic growth of a sixfold product difference set
{title2=$|3SS-3SS|\geq\tfrac12\min(p,|S|^2)$}
For $S\subseteq\mathbb F_p$ with $|S|\geq2$, put $R=(S-S)/(S-S)$ using nonzero denominators. If $R$ is proper, some $r\in R+1$ lies outside $R$, since invariance under addition of one would give all of the <prime field>. The map $(x,y)\mapsto x+ry$ on $S^2$ is injective. Clearing the denominator in a representation of $r-1$ puts a dilate of its image inside $3SS-3SS$. If $R$ is the full field, averaging the collision count of $S+rS$ gives some $r$ with $|S+rS|\geq\frac12\min(p,|S|^2)$. Clearing its denominator puts a dilate inside $2SS-2SS\subseteq3SS-3SS$. Here $SS$ is a <product set>, and the prefactors denote repeated <sumsets>, not <scalar> multiplication.
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